\section{Chronology} 
\footnote{Based on Cosmochemistry, Harry Y. McSween, Jr. and Gary R. Huss,
Cambridge 2010.}
\label{appendix:chronology}

Determining the ages of events and placing them in chronological order 
constitute one of the major areas of research in cosmochemistry. There are
several different approaches to determine the timing of events. 
Among them radiometric age dating is a powerful chronological tool. 

The number of decays in a given period of time is proportional to the
number of atoms present:
%
\begin{equation}
-\frac{dN}{dt} = \lambda N
\label{eq:decay}
\end{equation}
%
where $\lambda$ represents the probability that an atom will decay within
a stated period of time and $\lambda N$ gives the rate of decay at a time.
The numerical value of $\lambda$ is unique for each radionuclide and is 
expressed in units of reciprocal time. If we integrate Eq. \ref{eq:decay} we
get the remaining number of atoms at time $t$:
%
\begin{equation}
N = N_0 e^{-\lambda t}
\end{equation}
%
where $N_0$ gives the starting number of atoms at time $t = 0$.

The number of daughter atoms ($D^*$) produced by the decay of the parent isotope
at any given time is
%
\begin{equation}
D^* = N_0 - N = N ( e^{\lambda t} - 1 )
\end{equation}
In most natural systems, the number of atoms of the daughter nuclide ($D$)
consists of initial atoms already present in the system ($D_0$) plus those
resulting from radioactive decay ($D^*$):
%
\begin{equation}
D = D_0 + D^* = D_0 + N(d^{\lambda t} - 1)
\end{equation}
%
Both $D$ and $N$ are measurable quantities and $D_0$ is a constant whose value
can either be assumed or calculated from other data. Solving the equation
above for $t$ gives the data:
%
\begin{equation}
t = \frac 1 \lambda \ln \left[ \frac{D-D_0}{N} + 1 \right]
\end{equation}

There are several requirements that need to be met to make sure that the
radiometric date provides a valid age.
%
\begin{enumerate}
  \item A specific event is required that homogenized the isotopic compositions
        of the parent and daughter element. It assures that additions to the
        amount of daughter isotope reflect only parent element
        and the time passed.
  \item The system, e.g. a rock or mineral, must remain closed.
  \item The decay constant ($\lambda$) must remain constant over the age of
        the solar system and the galaxy, and be accurately known.
  \item The initial abundance of the daughter element $D_0$
        is possible to assign with a realistic value. 
\end{enumerate}

With the formalism discussed above, one can study a wide variety of systems
to produce a radiometric date.

%----------------------------------------------------------------------------
\subsection{Half-life}\label{halflife}
%----------------------------------------------------------------------------

The rate of decay of a radionuclide is often described in terms of its
half-life. The half-life ($T_{1/2}$) is the time required for one-half of 
a given number of radioactive atoms to decay. So
%
\begin{equation}
\frac 1 2 N_0 = N_0 e^{-\lambda T_{1/2}}
\end{equation}
%
\begin{equation}
T_{1/2} = \frac {\ln 2}{\lambda} = \frac{0.693}{\lambda}
\end{equation}

Note that the half life is smaller than the mean life $\tau$:
%
\begin{equation}
\tau = \frac 1 \lambda
\end{equation}

%----------------------------------------------------------------------------
\subsection{Long-lived Radionuclides}\label{long}
%----------------------------------------------------------------------------

Long-lived radionuclides are those with half-lives long enough that still
a significant fraction of the original atoms in the early solar system
are present today. Long-lived radionuclides can give absolute agaes of events
back to the formation of the solar system. 
The U-Th-Pb system is a typical long-live radionuclide chronometer.

%----------------------------------------------------------------------------
\subsection{Short-lived Radionuclides}\label{short}
%----------------------------------------------------------------------------

The short-lived radio-nuclide $^{26}$Al (half life ~0.73 million years, decay
to $^{26}$Mg) was found in CAIs.
The ratio of the daughter isotope to a stable isotope of the same element
($^{26}$Mg/$^{24}$Mg in Fig.\ref{fig:al_mg}) 
increases as the radioactive isotope decays.
Because of its short half life, the existence of $^{26}$Al in CAIs is 
the evidence that it was alive in the early solar system. And due to the 
short time scale, it provides a high resolution way to study the early history
of the solar system.

\begin{equation}
\frac{\mg{26}}{\mg{24}} =
  \frac{\mg{26}_0}{\mg{24}} +
  \frac{\mg{26}^*}{\mg{24}} =
  \frac{\mg{26}_0}{\mg{24}} +
  \frac{\al{26}}{\al{27}} \frac{\al{27}}{\mg{24}}
\end{equation}
  
An interesting phenomenon is that
$^{26}$Al and neutron-rich iron-group anomalies seem ``mutually exclusive'' 
(Andy Davis's words).
%--and see Hinton et al. 
In FUN CAIs there is little $^{26}$Al, but it shows excesses and deficits in
n-rich iron-group isotopes, which is not found in regular CAIs.
Then the question comes out:
Did the FUN CAIs form before $^{26}$Al
came into the Solar System, or was the 26Al heterogeneously distributed?

\begin{figure}[ht!]
\centering
\includegraphics[width=\figuresize\textwidth]{figures/al_mg}
\caption{
Magnesium isotopic ratios measured in different minerals with different 
ratios of aluminum to magnesium from a refractory inclusion in the 
meteorite Allende. Magnesium shows excesses in the isotope $^{26}$Mg that 
are correlated with the aluminum/magnesium ratio, indicating that the 
$^{26}$Mg excesses originated from the decay of the radioactive isotope 
$^{26}$Al. 
This finding is evidence for the initial presence of $^{26}$Al in early solar 
system objects. 
\protect\cite{1976GeoRL...3...41L}
}
\label{fig:al_mg}
\end{figure}

